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Class 12 th Mathematics chapter wise test

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224 views72 pages

Maths Q&B

Class 12 th Mathematics chapter wise test

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Shreyash Jadhao
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© © All Rights Reserved
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, R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-01 CLAS - XI MATHEMATICS CH-01 Relation and function 1. ARelation R:A-A is said to be Reflexive if --------- for everyae Awhere Aisnon [1] empty set, A Relation R:A>A is said to be Symmetric if ARelation R:ADA is said to be Transitive if ~ Define universal relation? Give example. What is trivial relation? Let T be the set of all triangles in a plane with Ra relation in T given by R= {(T1,T2):T1 is congruent to T2}. Show that R is an equivalence relation, Show that the relation R in the set Z of integers given by R = {{@, b) : 2 divides a-b}. is equivalence relation. Let L be the set ofall lines in plane and R be the relation in L define if R= {(h, Lz): Lais 1 to La} Show that R is symmetric but neither reflexive nor transitive. 9. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as (4 R= {(a,b): b =a+1)} is reflexive, symmetric or transitive. Let A= R - {3}and B = R-{1}. Consider the function f :A = B defined by Is f one-one and onto? justify your answer. \x Let L be the set of all lines in xy plane and R be the relation in L. define as R= (L112): La |] La} Show then R is on equivalence relation. Find the set of all lines related to the line y=2x+4. Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer , R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-05 CLASS - XII MATHEMATICS. CH-05 Relations and Functions show that a one - one function f: {1, 2,3} {1, 2,3} must be onto. If f(x) is an invertible function, find the inverse of f(x) = —— Let S = {1, 2, 3} Determine whether the function f: S > S defined as below have inverse. f={ (1,2) (2,1) (3 1)} Find fog f(x) = 8x3, g(x) =x"/3 If f: R > R be given by f(x) = (3—x°) , find fof (x) 6. &R> Rbe defined as f(x) = 3x check whether the function is one - one onto or other [4] 7. Show that the relation R defined by (a, b) R(c, d) =>a+b=b+con the set N xNis an equivalence relation. 4) 8. Let L be the set of alll lines in Xy plane and R be the relation in L define as [4] (La, L2): La || Lz} Show then R is on equivalence relation. Find the set ofall lines related to the line y=2x+4, ..C.M ofaand b. find [6] 9, Let * be the binary operation on H given by a* b= (a) 20* 16 (b) Is * commutative (0) Is* associative (d) Find the identity of * in N 10. Ifthe function f: R> Ris given by f(x) = =** and g: R > Ris given by g(x) = 2x-3, Find ()fog (i) gof. Is Ft = [6] Show that the function f:R {xe R: -I<.x S defined as below have inverse. F={G, 3) (3,2) (2, DP Find gof where f(x) = 8x3, g(x) = x!/3 Let f, gand h be function from R +R. Show that (fg) oh= (foh). (goh) Let f: R > R be define as f(x) = x* check whether the given function is one-one onto, [4] or other Show that the relation R defined in the set A of all triangles as {(7,,7,):7) is similar to T2}, is an equivalence relation, Consider three right angle triangles T1 with sides 3, 4, 5. T2 with sides 5, 12, 13 and Ts with sides 6, 8, 10. Which triangles among T1, Ta and Ts are related? |. Determine which of the following operation on the set N are associative and, which are commutative. (@a*b=tforallaben — (b)atb= *" forallabyen Let A and B be two sets. Show that f: A xB > B XA such that f(a, b) = (b, a) is a bijective function. [6] n+, if n is even. Show [6] 10. Let f: W > W be defined as f(n)=n-1, ifn is oddand f (1 that / is invertible. Find the inverse of /. Here, W is the set of all whole numbers. Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-03 CLASS - XII MATHEMATICS CH-01 Relations and Functions Show that function f: N > N, given by f(x) = 2x, is one - one. a Let $ = {1, 2, 3} Determine whether the function f: $ > S defined as below have inverse, f={(1, 1), (2,2), (3,3)} (1) Find got f(x) = |x|, g(8) = [5x -2| (1) Consider f: {1, 2, 3} > {a, b, c} given by f(1) =a, f(2) = b and f(3) =c find f4 and show that (F4)4 =f Q) If f(a) = x + 7 and g(x) = x- 7, V xe R find (fog) (7) ish State whether the function is one - one, onto or bijective f: R > R defined by f(x) = 3 - 4x [4] Show that the relation R in the set of all books in a library ofa collage given by R={O0y) ixandy have same no of pages}, isan equivalence relation, Let f:R > Rbe f (x) = 2x+ Land g: R > Rbe g(x) = x? - 2 find (i) gof (ii) fog 9. Let* bea binary operation, Given by a*b=a-b+ab 4] a, (a) Commutative (b) Associative Let A = R - {3} and B = R- {1}. Consider the function of f A > B defined by f(x) = ‘one — one and onto, Let A = Nx Nand * be the binary operation on A defined by (a,h)*(c,d)=(a+c,b+d) Show (6) that * is commutative and associative. Find the identity element for * on A, if any. re a eT ae a Deere eae Rimmer Bice eee JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS - XI] MATHEMATICS Relations and functions Prove that the function f: R > R, given by f(x) = 2x, is one - one (a) (1) Let S = {1, 2, 3) Determine whether the function f: S > S defined as below have inverse, f= {(1, 2), (2, 1), (8, 1)} Find gof f(x) = [x], g(x) = [5x + 1] res] State whether the function is one ~ one, onto or bijective f: R > R defined by f(x) = 1+ x? [2] Let f, gand h be function from R to R show that (f+ g) oh = foh + goh Ifa*b=a+3b?, then find 2*4 (2) Show that the relation in the set R of real no. defined R = {(a, b) : a< b3}, is reflexive nor symmetric nor transitive. [4] ifn is odd for allne N 8. Letf:N > N bedefined by f(x) = Examine whether the function fis onto, one - one or bijective Let A = NN and* be the binary operation on A define by (a, b) * (Gd) =(a+cb +4) Show that * is commutative and associative. 10. Show that if f. an-{3} aah} iSefining by {x)= +“ atid g: is define by {3) 3 sx—7 ee | sf Ix+4 3 (x) = + then fog =I, and gof=Ip when 4=r—[3\. per— 2}: IN oo=% for all xe A, x3 3 3 In(X) = x, for all xe B are called identify function on set A and B respectively [6] 11. Consider f:R, ->[-5,~] given by f(x) =9x° +6x-5. Show that / is invertible with ((Jy+6)-1 ry) Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X Cc TEST PAPER-03 ASS ATHEMATICS CH-02 Inverse Trigonometric Functions 1 Find the value of cot(tan"' a+ cota) t 2. Find the principal value of cos" | 3. Find the value oftan” | tan| 2 || (1) 1 Prove that tan"! Vx = —cos tan} tan= | es Prove 2tan 1 acosx~bsin x) 7. Simplify tan? S£08* SY (bcosx+asin x ) Ifa>b>c>0 prove that Madd) (bet cot" + cot ab) Prove thattan’ i : If sin(sin”' —+ cos” x) = 1 find x [6] re a eT ae a Ee Lee er TL Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS - XII MATHEMATI CH-02 Inverse Trigonometric Functions Find the principal value of cot” 2. Find the value of cos oo 35)], Find the value of sin| (ab+1) 4. Prove that cot” amb } \b=e) sin(tan“ x) =? [2] . -if_cosx_) Explore tan“'| °° in the simplest form, 4 12 4ms Show thatsin“!7— tos! tan gis tnx Prove that cot| =i 4) Write in simplest form that tan” +2tan” or Prove that 2sin" re a eT ae a Ee Lee er TL Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-01 CLASS - XII MATHEMATI CH-02 Inverse Trigonometric Functions Find the value of sin (sin”' a+cos“' a) Find the value of sin+ { sin . : ) Find the value of tan” Y3 ~cor™'(~y/3) Find the principal value of sin? evaluate Wy) \xty) pI Find the value of tan”'(1) + cos" | 3 8 Show that sin“! 2~sin“* = cos Prove that tan” x+ tan So oe Prove that tan"? tan! tan“! 5-+tan = = =(3 | Jor cos” | cosx +Esin x sinx-+¢os.x ) 10. Simplify sin” re a eT ae a Ee Lee er TL Rimmer 3 R.K.MALIK S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X EST PAPER-05 CLASS - XII MATHEMATICS CBSE CH-01 Inverse Trigonometric Functions Find the principal value of sec Find the value of “cos {cos L Find the value of cot” Find tant (, fy) 5. Find the value of tan”) 2 cos) 2sin xo If tan! + tan x2 Show that sin” Solve tan re a eT ae a Ee Lee er TL Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-04 CLASS - XI! MATHEMATICS CH-02 Inverse Trigonometric Functions Find the value of cos(sec' x +-cosec"'x) ecw 2) 1) 2. Find the principal value ofc 3. Find the value ofsin” sin] = Prove that tan” (3a*x—x? ) 5. tan” | > _ ? 1 \P—3ax7 ) a 1 Find the value tan—| sin Solve tan 2x-+tan 5 De Prove that If cos” 2xy yaa oosr+% =sin?a ab 5 Prove that re a eT ae a Ee Lee er TL Rimmer , R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV,), MEDICAL « BOARD, NDA, IX & X CBSE TEST PAPER-03 CLASS - XII MATHEMATICS CH-08 Application of Integrals Note : Each Question carries 6 marks. Find the area bounded by the curves (x ~ 1) 2+ y2= Land x?+y?= 1. Find the area of the region bounded by the parabolas y? = 4ax and x? = day, a > 0. Find the area of the region bounded by the curves y = 2+ 2, y=*,x=0 and x= 3. Find the area of the region {(x.y): x Find the area bounded by the curves {(x,y) 217+" $ 2ar, y* > ax.a > 0,x>0,y>0} Using integration, find the area of region bounded by the triangle whose vertices are (-1,0);(4,3) and (3, 2) sisx+y} Find the area of the region:{(x,y):x°++y" $1<.x+y} 8. Draw a rough sketch of the region {(x,):.y° $3x,3x°+3)" =16}and find the area enclosed by the region using method of integration, Using integration find the area of the triangular/ region whose side have the equations y = 2x + 1,y = 3x + t,and x= 4, Calculate the area of the region enclosed between eh circles: x2 + y? = 16 and (x + 4)? +y?=16, Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer 5 R.K.MALIK S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS - XII MATHEMATICS CH-08 Application of Integrals Note : Each Question carries 6 marks. ‘The area between x= y? and x = 4 is divided into equal parts by the line x the value of a Find the area of the region bounded by the parabola y = Find the area of ellipse Find the area bounded by the curve x? = 4y and the line x = 4y ~ 2. Find the area of the region bounded by the curve y? = 4x and the line x = 3 Find the area between the curve y= |x + 3], the x-axis and the lines x = -6 and x = 0 Find the Area lying in the first quadrant and bounded by the circle x? + y? = 4 and the lines x= 0 and x = 2. Find the Area of the region bounded by the curve y’ x, y~ axisand theline y = Find the area enclosed between the curve y = x° and the line y = x. Find the area of the circle 4x?-+ 4y? = 9 which is interior to the parabola y? Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Ee Lee er TL Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-O1 CLA‘ XII MATHEMATICS CH-08 Application of Integrals Note : Each Question carries 6 marks. =x and the lines x= 1,x=4and Find the area of the region bounded by the curve y’ x-axis. 2, Find the area of the region bounded by y? = 9x, x =2, x = 4 and the x ~ axis in the first, quadrant. Find the area of the region bounded by the parabola y + 1and the linesy=%, x=Oand x= 2. Find area of the region bounded x? = 4y, y = 2, y = 4 and the y ~ axis in the first quadrant Find the area of the region bounded by the ellipse: * yay 9 Find the area of the region bounded by the ellips Prove the area of a circle of radius ris zr? square units. Find the area of the region in the first quadrant enclosed by x - axigand x =/3y by the circle x2 + y? Draw the graph of the curve. y= and find the area bounded by this curve and the coordinate axis. Find the area of the smaller part of the circle x? + y? = a? cut off by the line x= Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer aes els JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X T PAPER-05 CLASS - XIl MATHEMATICS CH-07 Integrals x) + xcos x4 tan’ x+1)dx Show that [° /(2) e(ode=2[" fod 21 If f(a) = f(a—x)and g(x) + g(@-x) **(2logsin x —log sin 2x) dr 2+sin 2x 2esin2x 7 La 1 Tr cose fran *| x8in(aex) | ax 136 413r—10 re a eT ae a Deere eae Rimmer Ban eit JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-04 CLASS - XII MATHEMATIC CH-07 Integrals +1)(+ logs) dx ii] 0s 2x 008 2a he 1, (cos x cos ar . 1) 5. fe*| tant e+ be [2] Jog (I+ tan x)dx [est + |e 21+ |x=3iJae xdt os" x+5" sin x Find its sum of limit ["(x-+e7*)dx [6] re a eT ae a Deere eae Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-03 CLASS - XII MATHEMATICS, CH-07 Integrals 0x" +10" logelO x+10' os2x-+2sin?x eS cc uy cos" x [ie sin’ x de prs de fn + Vtanx 2) sin x cosy Se tx i cos" x-Fsin’ x a ng hi cos x+4sin™ x logsin x de re a eT ae a Ee Lee er TL Rimmer Became JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-03 CLASS - XII MATHEMATICS CH-09 Differential Equations 1. Find order and degree. fy (dy) (a y dy\ 2) (2 ssn 2)r<0 (ae) “Cae at 2. Verify that the function is a solution of the corresponding diff eq, [2] y lex lx 3, Form the differential equation representing the family of curves given by (x: (4) a)? + 2y?= a2, where a is an arbitrary constant. Form the diff. equation of the family of circles in the second quadrant and totiching the coordinate axes. Solve the diff eq. (verexey® sy=Lwhenx =0 1) Lar; y=0 when x2 4 Solve x Solve | 1+e" dete? | Solve x Sin’ de-+x dy=0 :y=2/4, when x= Solve the eq, (1+?) de= (tan? y—x)dy 10. Solve the diff eq. re a eT ae a Ee Lee er TL Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS - XII MATHEMATICS. CH-09 Differential Equations Find the order and degree 3rd 8-0 a ds) at) Verify that the function is a solution of the corresponding diff eq xty=tan"ysy?y! ty? +1 y 0 Form the differential equation representing the family of ellipses having foci on x-axis and centre at the origin. Form the diff. eq of the family of circles touching the x - axis at or Solve the diff eg ec'y dy =0 © tany dx + (I-e*) (ay) Solve Cos {“%)=a,; y=1/when=0 Solve. (x7 -y?}dx+2xy dy = 0 Solve 4X Cos Find the particulars solution of diff.. equation. “own de (14s? Jay 2a Find the particular solution of diff. equation tay Cotx=0 Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Ee Lee er TL Rimmer R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-O1 CLASS - XII MATHEMATICS CH-09 Differential Equations Find the order and degree. yrty*te" 0 Verify that the functions is a sol of the corresponding diff. req xsinx cay" =y+xqx 3. Form the differential equation of the family of hyper bolas having foci On x-axis [) and center at origin Form the differential equation of the family of circles having centre on y-axis and radius 3 units, Solve the diff equ. sec’x.tan y dx+sec” y tan x dy =0 Solve the diff eq. y log y dx -x dy=0 6 solve x eos (2)%= yoos g,\ Solve i) 2yetde+(y and x=0when Find the general sol, of the diff’ eq, de 10, Find the particular sol of the diff. eq [6] a A 2s yootx=2x+2" cote de Given that y = 0 when x = 4 Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Ee Lee er TL Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-05 CLASS - XII MATHEMATICS CH-08 Application of Integrals Note : Each Question carries 6 marks. 1. Find the area of the smaller region bounded by the ellipse | and the line 2. Find the area oftthe region enclosed by the parabola x theline y =x +2 and the x-axis. Using method of integration, find the area bounded by the curve |x| + |y| =. i Using method of integration find the area of the triangle ABC, coordinates of whos Find area bounded by curves {(x.y): y 22° and p vertices are A (2, 0), B(4,5) and € (6,3). Using method of integration, find the area of the region bounded by lines: , Bx - 2y = 6 axty and x~ 3y +5=0. 7. Find thearea of two regions { Find the area of the circle x2= y? = 15 exterior to the parabola y? = 6x (xy) :37 <4x,4x° 44y? 9} Find the area bounded by the y - axis, y= cosx andy = sinx, 0< x Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line and the circle x? + y? = 32. Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-04 CLASS - XII MATHEMATICS CH-08 Application of Integrals Note : Each Question carries 6 marks. Using integration, find the area of the region given below {(s»):05 pS +105 ysx+L0sxs2}, Compute the area bounded by the lines x # 2y=2,y-x= Land 2x+y =7. Find Smaller area enclosed by the circle x? + y? = 4 and the lines x + y = 2. Find the area between the curvesy = xand y =x%, 3a Sketch the graph of y = |x +3] and evaluate [| Find the area bounded by the curve y = sinx between x = 0 andx = 2 Find the area enclosed by the parabola y? = 4ax and the line y = mx. Find the area of the region {(x,y):0< p< (x? +1),0¢ y<(x+I),0 [4] (nx e y= logtan{ £+=) show that 2 seex-=0 il re a eT ae a Ee Lee er TL Rimmer 5 R.K.MALIK S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-03 CLASS - XII MATHEMATICS CH-05 Continuity & differentiability Find the value of K so that function is continuous at the given value. Kxtl if. xa f=} cox if x>a 2. Differentiate y 3. Find 2 isin? y+e0sxy= i] ae 4. ind 2 yyw ae a i ag dy Find = when x =a(@-sin@), y=a(1+cos@) dy _ cay 6. I y=3e% + 2c Prove that £2 -s2'46y=0 il ac ds * Show that (I=) 14 If (x-a)’ +(9—b)* =e? Prove ~“1_ is a constant independent of a&b. _/ [4] Find %, ity sin'x de 10. y=(sinx—cosx)"*"" re a eT ae a Ee Lee er TL Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS XII MATHEMATICS CH-05 Continui Find all points of discontinuity if & differentiability x|43, if if if f(x) =} 2x, 6x42, and are at right angles. 3 2p 6 Find the shortest distance between the lines whose vector equation are [4] r=(-ni+0= F=(s4 Dit 2s) 7-2s+Dk Find x such that four points A(3,2,1) B(4x5)(4,2-2) and D coplanar. (6.5,-1are a, Find the angle between the two planes 2x +y-2z=5 and 3x-6y -2z= 7using _ [4] vector method. Find the equation of the point where the line through the points A(3,4,1) and B(5,1,6) crosses the XY plane. 40, Prove that if a plane has the intercepts ajb,c is at a distance of p units from [6] the origin then Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-01 CLASS - XII MATHEMATICS, CH-05 Continuity & differentiability A Find the values of K so that the function fis continues at the given value of x. Keosx a aS ip xe ax 2 5 its Differentiate the function x*** +(sin.x)"*" [4] Weave" y *T show that 2 — aa x Ify = (tan-tx)? show that (x2 + 1)? yo # 2x (x?+ y= 2 Verify Rolle’s Theorem for the function y = x? +2, [-2,2] 6. Differentiate sin") = Differentiate sin2x w.r. to e% if xJTep + yi =0 prove that de (rexy dy _cos'(a+y) ix sina 9. Ifcosy = x cos (a+ y) prove that “ 10. 1fx=a (cost + tsint) y=a(sint-teost) find 2 & re a eT ae a Ee Lee er TL Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-05 CLASS - XII MATHEMATICS CH-04 Determinants then show that |24]=4]4] 0] 1 1, If A= 4 A be a non — singular square matrix of order 3 x3. Then |adj | is equal to If A is an invertible matrix of order 2, then det is equal (A) to B=[-1] find det B A 5. Show that using properties of det.|y | y* zx]=(e=y)(y-zz xy + y2+ 21) [4] m4 xy la @ By 6 |B B y+a\=(B-y)\(y>a)(a=f)(a+ B+) [4] y\ 7 arb ind values of K if area of triangle is 35 square. Unit and vertices are (2, -6), (5, 4), (K, 4) Using cofactors of elements of second row, evaluate A Show that A? — SA-+ 71 = 0. Hence find A” If A= ‘The cost of 4kg onion, 31 wheat and 2kg rice is Rs, 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer , R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-04 CLASS - XI MATHEMATICS, CH-04 Determinants than x is equal to is singular or not Ita 11 ht h 1 Ate} Show that using properties of det. =abe+be+eatab 14" 6. Ifx.y, zare different and A=|y y* — 1+y’]=Othen show that 1 +xyz=0 (4) 12) Zz Find the equation of the line joining A (1, 30 and B (0, 0) using det. Find K if D (K, 0) is a point such then area of A ABC is 3 square unit (21 8, Show that the matrix 4 = | satisfies the equation A? 4A+ 1=0. Using this equation, find At Solve by matrix method 3x-2y +32=8 axty-z=1 4x—3y+22=4 The sum of three no, is 6. If we multiply third no, by 3 and add second no. to it, wwe get II. By adding first and third no. we get double of the second no. represent it algebraically and find the no. using matrix method. Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV,), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-03 CLASS - XII MATHEMATICS CH-13 Probability 1. Find the probability distribution of number of doublets in three throws of a pair of [2] dice. Let X denote the no of hours you study during a randomly selectee school day. The probability that X can take the values x, has the following form where K is some unknown constant O.1ifx=0 kx if'x= 1, or2 KG -x)ifx=30r4 0, othenvise (a) Find the value of K (b) Whatis the probability that you study at least two hours. Exactly two hours? At most 2 hr. [4] p(z=x) Find the variance of the number obtained on a throw of an unbiased die. Two cards are drawn simultaneously (or successively without replacement) for a well shuffled of 52 cards. Find the mean, variance and standard deviation of the number of kings. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs, In a meeting 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take x = 0 if he opposed and x = 1 if he is in favour. Find E (x) and var (x). Aand B throw a die alternatively till one of them gets a their respective probabilities of winning if A starts first and win the game. Find Find the mean of the Binomial distribution B { 4, 3) 2 9. Ifa leap year is ‘Tuesdays? elected at random, what is the chance that it will contain 53. [2] Bag | contain 3 red and 4 black balls and bag II contain 4 red and 5 black balls. One ball is transferred from Bag I to Bag Il and then a ball is drawn from Bag Il. The ball so drawn is fund to be red in colour. Find the probability that the transferee ball is black. re a eT ae a Deere eae Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS - XII MATHEMATICS CH-13 Probability Given three identical boxes 1, I and Ill each containing two coins. In box-1 both coins are gold coins, in box-Hl, both are silver coin§ and in the box-III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold. Suppose that the reliability of a HIV test is specified as Follows of people having HIV, 90% of the test detect the disease but 10% go undetected of people free of HIV, 99% of the test are Judged HIV ~ tive but 1% are diagnosed as showing HIV ‘ive From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports himvher is HIV tive \what is the probability that the person actually has HIV In a factory which manufactures bolts, machines. A, B and C manufacture respectively 25%, 35% and 40% of the bolts, Of their output 5,4 and 2 percent are respectively defective bolts. A bolt is drown at random from the product and js found to be defective. What is the probability that it is ma wufactured by the machine B, ‘A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by other mean of 1 transport are respectively 3 t z A the probsbitities that Ke will be 1 1 lateare 7, >, and 75 ifhe comes by train, bus and scooter respectively, but he comes by other means of transport, that he will not the late. When he arrives he is late, What is the probability that he comes by train. ‘Aman is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. re a eT ae a Deere eae Rimmer DOWNLOADED FROM NEWTON CLASSES.NET In answering a question on a multiple choice test a student either knows the 3 answer or guesses Let “be the probability that he knows the answer and % be the probability he guesses. Assuming that a student who guesses at the answer will be correct with probability + What is the probability that the student knows the answer given that he answered it correctly. A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e if a healthy person is test then with probability 0.005 the test will imply he has the disease) If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive. ‘An insurance company insured 2000 scooter drivers, 4000, car drivers and 6000 truck drivers. The probability of accidents is 0.01, 0.03 and 0.15 respectively. One of the insured persons meet with an accident what is the probability that he is scooter driver. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card beinga diamond, Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3, 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one had, what is the probability that she threw 1, 2, 3 or 4 with the die? rr a ea DeLee ey Rimmer DOWNLOADED FROM NEWTON CLASSES.NET CBSE TEST PAPER-02 CLASS - XII MATHEMATICS CH-13 Probability {answers} Ans 01, let E;, Ez and Es be the events that boxes 1, and III are chosen, P(E) =P (E)=P E) let A be the event the coin drawn is of gold. P(AIE,) = PG) PAE) P (E,) P(AIE,) + P (E,)P (AIE,) + P (E,) P (AIE,) PIA) let E denote the event that the person selected is actually haying HIV and A the event that the person’s HIV test is diagnosed as + tive let £” not having HIV. ou P(E) =0.1% = — = 0.001 150, P E)=1-PE)=0.999 90. P(AIE)= 90% = > =0.9 A 100 P(AE)=1% == 001 100 P(E) P(AIE) P(E) P(AIE)+ PE) PAE) = 0.083 P(EIA)= Ans 03. let. By =bolt is manufactures by A B2 = bolt is manufactured by B B; = bolt is manufactured by C let E bolt is defective re a eT ae a Ee Lee er TL Rimmer Ans 04. Ans 06 DOWNLOADED FROM NEWTON CLASSES.NET P(B,)=25% = 02: P (B:)=0.35 P (Bs) =0.40 P (EIB) = 5% P (E)B2) = 0.04 P (EBs) = 0.02 PB.) EIB.) PEE 28 BS ae P(B)P EB) +P (B,) (PEIB.) + PB) PEB,) oO let E be the event that the doctor visits the patient late and let Ty, Ta, Ts, Ts, be the event that the doctor comes by train, bus, scooter and other means of Transport respectively P(ny= >. pcr = 4, PCL)* 4, Peay = 0 10 P (Bi, = +, p (EIT) = +. PET) = 4. PENT) =0 a 3 2 P(T)P (EIT) PCIE)= - P (T,)P (EIT) + PCT, )P ((E[T,) + PCT) P (E/T) + P (T,) P (EIT, ) let E be the event that the man reports that six occurs in the throwing of the dice and let 8} be the event that six occurs and S, be the event six does not occur. P(BS)= >, Pes,)=1 be ) P(S,) P(E)S,) P(S)P ES) +P SP ES.) 3 P(S/B= 8 E; : the student knows the answer Ep : the student guesses the answer A the answer is correct 1 P(AIE\)=1, P (AVE) = ( c ; re a eT ae a Deere eae Rimmer Ans 07. Ans 08. DOWNLOADED FROM NEWTON CLASSES.NET P(E) P (ALE) P (E)P (A/E,) + P (E,)P (ATE) E; : the person has the disease E, : the person is healthy A sHestis positive 9 10 10 P(E)=01 +, P(E.) 16 99 pe.) = 0.005 P(AE:)= joo” ? WF») 5 000 P(E) P(A YP (AIE,)-+P (Es) PAE) 9 1 x BA Ted B a xt sy 100 * 10 *1000*10 Ey: Insured person is a Scooter driver E2: Insured person ia car driver Es : Insured person is a truck driver 2000 2006 Pe) 2000 2000 1 2000+ 4000+ 6000 12000 6 PE P(E)= re a eT ae a Ee Lee er TL Rimmer Ans 09, DOWNLOADED FROM NEWTON CLASSES.NET let A Insured person meets with an accident P(A/E,)=0.01 P (AE,) = 00: (AE) = 003 = = P(AB,)=0.15 = 100 P(E) P(A.) P (E,/A) = Oe P(E) P (al P(E) PAV EI : lost card is diamond E2 : lost card is not diamond let A; two cards drawn from the remaining pack are diamonds. P(E)= Bot pe)-2 3 4 12x11 51x30 3x12 51x50 P(E) P(AE,) P (AI) P (AIE,)= P(E/A)= E, : 1, 2,3, 4 is shown on dice Ep: 5 or 6 is shown on dice 4 Pe)= 4-2 Pe)= 6 3 y let A exactly one head shown up P(AE,)=+, P(AE,)=3 2 enn P(E) P(A) S P(E, /A) = ent P (E)P(AE,)+P(E,)P(AE,) 1 re a eT ae a Ee Lee er TL Rimmer DOWNLOADED FROM NEWTON CLASSES.NET CBSE TEST PAPER-03 CLASS - XII MATHEMATICS CH-13 Probability Find the probability distribution of number of doublets in three throws of a pair of [2] dice. Let X denote the no of hours you study during a randomily selectee school day. The [4] probability that X can take the values x, has the following form where K is some unknown constant O.1ifx=0 kx if'x= 1, or2 KG -x)ifx=30r4 0, othenvise (a) Find the value of K (b) Whatis the probability that you study at least two hours. Exactly two hours? At most 2 hr. p(z=x) Find the variance of the number obtained on a throw of an unbiased die. Two cards are drawn simultaneously (or successively without replacement) for a well shuffled of 52 cards. Find the mean, variance and standard deviation of the number of kings. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs, In a meeting 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take x = 0 if he opposed and x = 1 if he isin favour. Find E (x) and var (x). Aand B throw a die alternatively till one of them gets a‘6’ and win the game. Find their respective probabilities of winning if A starts first: Find the mean of the Binomial distribution B { 4, 3) If a leap year is selected at random, what is the chance that it will contain 53 ‘Tuesdays? Bag | contain 3 red and 4 black balls and bag II contain 4 red and 5 black balls. One ball is transferred from Bag I to Bag Il and then a ball is drawn from Bag Il. The ball so drawn is fund to be red in colour. Find the probability that the transferee ball is black. re a eT ae a Deere eae Rimmer DOWNLOADED FROM NEWTON CLASSES.NET CBSE TEST PAPER-03 CLASS - XII MATHEMATICS (Probability) Topic: - Probability [ANSWERS] Let x denote the number of doublets x= 0, 1, 2,3 Probability of getting doublet = © = 366 Probability of not getting doublet = The probability distribution of x is ee |(0 1 P(x) [01 @Xpirl 0.1 +K+2K+2K+K K=0.15 (b) p (study atleast two hr) = p (x2 2) 2K +2K+K =5K 5 x0.15 =0.75 p (Study exactly two hr) =p (=2) =2K =2% 015 =03 p (Study et most two hr) = p(x < 2) =O01+K+2K =05 re a eT ae a Ee Lee er TL Rimmer 5 R.K.MALIK S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X T PAPER-03 CLASS - XII MATHEMATICS CH-10 Vectors 142) +3k 3. Find the direction ratios and the direction cosines of the vector , 2i Evaluate the product (34-55) (2a+7) @) +5)-2k Find axb if a=2i+j+3k, b= -8)B(S, 0, -2) and ©(11, 3, 7) are collinear, and [2] Show that the points A (1, find the ratio in which B divides AC. 7. Finda i) vector d which is L to both Gand 6 and é.d =15 44+ Let @ b=3/-2j+7k jE Let 6,5 and 2 be tvee vectors such that [i] =3,f]=4, []=5 and each one of them being to the sum of the other two, find |a+5-+¢ ah basi+2j-3) Find the angel between the vectors 4-+B and @~5 Find the sine of the angel between the vectors @=2)-J+3k 3) 42k Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Ee Lee er TL Rimmer 5 R.K.MALIK S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS - XII MATHEMATICS, CH-10 Vectors Is the measure of 10 Newton is scalar or vector. Write two different vectors having same magnitude. Find the direction ratios and the direction cosines of the vector 7 Find {a—5 |if {al If @=4743).42h b= 31+2k find |bx2al Consider two point P and Q with position vectors OP =3a~2h and OG =a+b - Find the positions vector of a point R which divides the line joining P and Q in the ratio 2:1 (i) internally (ii) externally. Show that the c with vectors points A, B, position —2a+3b+5¢, a+2b+3¢ and 7a—c respectively are collinear. Find a unit. vector 1 to each of the _ vectors +j+k, bai+2j+3k (a+5)and(a=b) where The scalar product of the vector i++ with a unit vector along the sum of vectors 2i+47-Sk, and 2i+2]+3h is equal the one. Find the value of A Find the area of the A with vertices A (1, 1,2) B (2,3, 5) and C (1, 5,5). Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-01 CLASS - XI! MATHEMATICS CH-10 Vector: Is the measure of 5 seconds is scalar or vector? Find the sum of the vectors, 2144) +k —6j-7k the direction cosines of the vector Find ratios and_the-direction p=2i-77 3h Find the angle between vectors a and b if [2] Vectors sand 8 be such that fi] = and nen xb ta unit vedo: Find angle betweena and 5 Find the unit vector in the direction of the sum of the vectors a=2i42}—Sk, b= 2+ j+3h 7. Show. that the points 4(2i-j+£).B(i-37 vertices of right angled triangle. Show. that the points 4(-2/+37+5k),B(7+2)+3k)and C(7i-k) are collinear, If a,B,¢ are unit vector such that ¢+4+¢=0 find the value of ab+bc+ca If a=23+2)+3k, b=-i4+2]7+h ,C=3i+7 are such that a+ Ah is Lio ¢ is then find the value of 4. Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer , R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-05 CLASS - XIl MATHEMATIC: CH-09 Differential Equations Write the order and degree of the diff equation y= Verify that the given Functions is a solution of the corresponding diff eq y=Cosx +e:y' +Sinx=0 Form a differential equation representing the given family of curve by elimination arbitrary Constants aandb. y= ae“ +be™ Form a differential equation representing the given family of curve by elimination arbitrary Constants aand b. y" =a(6°-x") Find the particular Solution of the diff, equation (Ie )dy+(I4¢y")e" dx =0- given thaty=1, when x=0 Solve the diff. eq de ex+1 Solve the diff. eq ify = bwhenx=1 Solve the following diffeq. (Burt y") dx + (?txy) dy =0 Solve the following diff. eq, (e+) S+2y=Ve +4 ae 10. solve the diff. eq, Bary tan x=Sinx ae Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer 5 R.K.MALIK S NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-04 CLASS - XI MATHEMATICS CH-09 Differential Equations 1 d a 1 Find order and depree. + sin(y*)=0 a) Verify that the function is a solution of the corresponding diff’ eq yor +2rtecy!2x2-0 Form a differential equation representing the given family of curve by eliminating arbitrary constants a and b. y "(axbx) Form a differential equation representing the given family of curve by eliminating arbitrary constants a and b. Y = e* (a Cos x + b Sin x) Solve the diff eq. Find the equation of the curve passing through the point | o, e } whose diff eg is Sin x Cosy dx-+ Cos x. Siny dy=0 =(y dx +x dy)x Cos Solve (x dy ~y dx) y'Sin { 8 Solve the diff eq, (J yet? dx= (ve +9") ay Find a particular solution ofthe diff eq. Given that y= 0 when x = 2/2 Solve the diff eq. Costx ® +y =tmx de Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer , R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-04 CLASS - XII MATHEMATICS, (Vectors & Three Dimensional Geometry) CH-IT Three Dimensional Geometry Find the direction cosines of the line passing through the wo points (2.4,-5) and (1,2,3). Find the equation of the plane with intercepts 2,3 and 4 om the x. y-and z axis respectively If the equations of a line AB is find the directions ratio of line parallel to AB Find the distance of a point (2,5,-3) from the plane r.(6i—3 +2k)=4 Find the angle b/w the line (2) I £43 x42 Find the shortest distance (742 )43h)+ AG—3}+ 2b and f= (41+ 5)-+68)4 u2i+3) +b) 7, Find the vector equation of a plane which is at a distance of 7 units from the [4] origin and normal to the vector 31+5)—6& 8, Find the Cartesian equation of plane (i+ j-&)= 9, Find the equation of the plane that contains the point (1,-1,2) and is Ltoeach of [6] the plane 2x#3y-27-5 and x+2y-32 = 8 40, Find the vector equation of the line passing through (1.2.3) and jjto the planes [6] 7@— +28) = Sand? Gi+iak Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-03 CLASS - XII MATHEMATICS, CH-11 Three Dimensional Geometry If aline has the direction ratios -18, 12, =4 then whatare its direction cosines Find the angle between the pair of line given by 2-aba(i+27+ 28) Fa 51-2} +uGh+2} +68) Prove that the points A(2,1,3) B(5, 0,5)and C(-4, 3,-1) are collinear find the distance between the lines |; and I: given by 2j-4k + A(i+3}+6k) h+wdi+3j+6k) 7 =3i43) Find the angle between lines Find the vector and Cartesian equations of the plane which passes through the point (5,2,-4) and .L to the line with direction ratios (2,3,-1) g, _ Find the Cartesian equation of the plane ic] PS—2Ni+ B=) +(25+ 08 9, Find the equation of the plane through the line of intersection of the planes [6] xty 42 Land 28+ 3y-+42=5 which is L of the plane xv + z= 0 40, Find the distance of the point (-1,-5.-10) from the point of intersection of the line [6] r= (2i—j+2h)+AGi+4j+2h) and the plane r.(i-j7+h)=5 Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS - XIt MATHEMA’ CH-11 Three Dimensional Geometry What is the direction ratios of the line segment joining P(x yi 21) and Q (xz y272) The Cartesian equation of a line is Find the vector equation for the line. 3 : 2) 3 show that the lines mg Hh are coplanar. 4, Find the shortest between the Ls and lp whose vectors equations are [2] i+ FAC F=2i+j-k+ Gi Find the angel between lines r= (2i-5j+h)+AGi+2) 46h) 72 (Th- ok) +2}-+ 26) : : a) 6 \show that the lines 2 =2 are perpendicular to each others Find the vector equations of the plane passing through the points R(2,5,3), QC-2-3,5) and T (5,3,-3) Find the Cartesian equation of the plane #(i4j-k) =2 Find the equation of the plane through the intersection of the planes 3x-y +2z-4=Oand x+y +z-2=0 and the point (2,2,1) to, If the points (1,1p) and (-3,0,1)be equidistant from the plane [6] 7.(3i+4j-12k)+13 =0, then find the value of p. Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer ’ R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-03 CLASS - XII MATHEMATICS CH-04 Determ Find value of x, if Find adj A for 0 sina — -cosa’ 3. Evaluate A=|sina 0 sinB 12 cosa sing 0 sty ytz zis 4, Without expanding, prove thatA=| 2 xy |=o Qi cei 1 5. If'matrix 4=|1 is singular, find x. (21 6. Ifa, b,cisin AP, and then finds the value of |x+3 [4] KH x 7. Find the area of A whose vertices are (3, 8) (-4, 2) and (5, 1) 4) 8 Show that, using properties if det. x7 1x. 1 a2 9. 4=|_ 7, Find the no, aand b such that A+ aA + bl=0 Hence find A Ia] uo Solve by matrix method x-ytz=4 2xty-32=0 xty+z=2 Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Ee Lee er TL Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-01 CLASS - XI] MATHEMATIC: CH-11 Three Dimensional Geometry Find the directions cosines of x,y and z axis. Find the vector equation for the line passing through the points (-1,0,2) and (3.4.6) Find the angle between the vector having direction ratios 3,4,5 and 4,-3, 5. 4, Find the vectorand Cartesian equation of the line through the point (5,2.-4) [2] and which is parallel to the vector 3/+27 Find the angle between the lines 6h) + (3-5) 4k) Find the shortest distance between the lines raG+2j+h+AG-F+0, ra(Qi=j-B+u 7+ 2k) direction cosines of the unit vector Lto the plane [4] Find the 761-3] -2k) +1= 0 passing through the origin. Find the angle between the two planes 3x - 6y + 22 = 7 and 2x + 2y- 2z=5 9, Find the coordinate where the line thorough (3,~4-5) and ((2,3,1) crosses [6] the plane 2x +y+2=7 Find the vector equation of the plane passing through the intersection of planes .0+j+k)=6 and £(2i+3}+4k 5 And the point (1,1,1) Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-02 CLASS - XII MATHEMATICS, CH-04 Determinants act 7 Sole oa +1 x1 Find minors and cofactors of all the elements of the det. l, 10218 3 36) 4 6 Evaluate | 17 sin10” costo’ 4. Show that sin80° _cos80" 0 4] Verify that «4, +4,,4y +44) latbx..ctdx pax la c p 6. Aslaxtb extd\ pxtql=(I-x*)b dq (4) u > \w uovow 34 7. asl ag argu that AB =] [4] Show that, using properties of determinants. 1+a°-b ab La*+b? 2b 2ab, =(I+a°+b? J) 2a Using matrices solve the following system of equation re a eT ae a Ee Lee er TL Rimmer DOWNLOADED FROM NEWTON CLASSES.NET. find AB and use this result in solving the [6] To solve the system of equations. x-y+2z=1 2y-32=1 3x-2y+4z=2 rr aa ee aS etree ee Rimmer , R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-05 CLASS - XII MATHEMATICS CH-10 Vectors Is the measure of 1000 em’ is scalar or vector. Write two different vectors having same direction. Find the direction ratios and the direction cosines of the vector 7 =i+ j-2h Find le between two veetors @ and b if|al=1, |b]=2 ab =1 5. 3 21 Find a vector in the direction of vector 4 that has magnitude 7 units 6 Ita =i+j+k, 6 = 7k find avector ¢ such that axc=h, and ac 7 A 4] 7. Find the Value of 2 so that the vectors 2/—4) + and 4i-8) + 2k are (i) parallel 4 (ii) perpendicular : Show that a t 7 2i+3)468) 5 I a=3i+j42k b 2) +4k find (i) Magnitude of axb i) Aunit vector a andl-B to both (iii) The cosine and cosine of the angle b/w the vectors @and 6 4] 10. a} ‘The vectors a and 5 = 217+ ple are mutually 1. Given lal =[}], find xandy Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Ee Lee er TL Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-01 CLASS - XII MATHEMATICS CH-04 Determinants Find values of x for which Abe a square matrix of order 3 x3, there |X| is equal to Evaluate A= Let || find al the possible value of x and y ifx and y are natural numbers 2} Fe 5. Find the equation of line joining (3, 1) and (9, 3) using determinants. [2] lx yz 6. Using cofactors of elements of third column, evaluate A=|l_ y 2x] [6] Iz xy 7. Show that, using properties of determinants. [6] la?+1 ac lab bPT be |=1ta° +87 40° lea | cb lwtzy oxy zx 8. xy (xtzye yz [6] xz ye (xtyy and B, 2x-3y + S2= 3x + 2y-4 xty-2z Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Ee Lee er TL Rimmer R.K.MALIK'S NEWTON CLASSES JEE (MAIN & ADV), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-04 CLASS - XII MATHEMATICS, CH-10 Vectors Is the measure of 30 m/s towards north is scalar or vector. Compute the magnitude of b=2)-7)-3k Find the direction ratios and the direction cosines of the vector r =i+27 Gls unit vectorand (¥—a)(x+a)=8, Then find |¥ Show that (a— x(a+b)=2(axb) a and b. Three vectors @,b and ¢satisfy the condition a+/+c=0 Evaluate the quanti ab-+bc+ca ule If with reference to the righthanded system of mutually 1. unit vectors i,j Rand k, @=3i —j, B=2i +] -3k thenexpress in the form B= +., where § is|| to @and B, isLto & If abandé be three vectors such that a+b+c=0 and {c]=7 find the angle between a and b. 9. Find the area of the ||gm whose adjacent sides are represented by the 4 2h, b=i-37 + 4k vectors, d= Find the vector joining the points P (2, 3, 0) and Q (-1, -2,-4) directed from P toQ. Also find direction ratio and direction cosine. Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Deere eae Rimmer 5 R.K.MALIKS NEWTON CLASSES JEE (MAIN & ADV.), MEDICAL + BOARD, NDA, IX & X CBSE TEST PAPER-05 CLASS - XII MATHEMATICS CH-03 Matrices Given an example of matrix A and B such that AB=0 but A #0, B #0 0 Tt -1} 2. Showthat 4=|-1 0 — 1|, is skewsymmetric matrix. 2 Loa oo , Prove that 4+" isa symmetric matrix 2] 24 5 6] es | Show that (34 Solve for x and y, given that then prove that A” [4] cos@ sind] fcosn@ sinné] -sin@ cos@ | sinn@ cosnd find x and y such that A?- xA + yl=0 : ay cos? = cosa sina] a-{ cos’ B cos B sing] leosa sina — sin’ cos sing sin’ B Show that AB isa zero matrixif a and f differ by an odd multiple of [4] vis Find the condition for which AB=0 IffQ)=x2-5x+7 and 4-[} Find X and Y, if 2x + 3y =| Office.: 606 , 6th Floor, Hariom Tower, Circular Road, Ranct Ee Lee er TL Rimmer

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